Proof of The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge

Each result cited below is universally quantified over the data in its own statement.

Throughout, $g$ is the zero function, so the term $g(x)$ in the operators of The Cutoff Hamilton-Jacobi-Bellman Operators of the Wick-Square Problem, with a General Counterterm is $0$.

Preliminary facts.

(P1) Real arithmetic. For a real $y$, $0\le y^{2}$: $|y|$ is $y$ or $-y$ and $0\le|y|$ (claim 1 of Properties of the Absolute Value in an Ordered Field), $(-y)(-y)=yy$ (claim 2 of Zero Products and Elementary Identities in a Field), so $y^{2}=|y|\,|y|\ge0$ by claim 5 of Elementary Arithmetic in an Ordered Field. For a nonnegative real $y$, $|y|=y$ (if $|y|=-y$ then $0\le-y$ and $0\le y$, so $y=0$, by claim 4 of Elementary Order Arithmetic in an Ordered Field). Two weak inequalities $s\le s'$, $r\le r'$ add to $s+r\le s'+r'$ by claims 3 and 2 of Elementary Arithmetic in an Ordered Field.

(P2) Constants. $0<\mu_{k}$ for every mode $k$, since $1\le\mu_{k}$ (The Wick-Square Problem on the Torus: Standing Notation §modes) and $0<1$ (claims 6 and 2 of Elementary Order Arithmetic in an Ordered Field); $\nu,\beta,\gamma$ are positive (The Wick-Square Problem on the Torus: Standing Notation §parameters), so $\gamma\ne0$, and products and inverses of positive numbers are positive (claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field). By The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §root, $0<q_{k}$ and $2q_{k}^{2}+(\gamma+2\mu_{k})q_{k}=\beta$. By The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §profile and claim 3 of Elementary Arithmetic in an Ordered Field, $q_{k}\le\frac{\beta}{2\mu_{k}}$. By The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §constants, $a_{k}=(\nu q_{k}-\beta c_{k})\gamma^{-1}$, so $\gamma a_{k}=\nu q_{k}-\beta c_{k}$; and $c_{k}=\frac{\nu}{2\mu_{k}}$ by The Free-Field Variances of the Fourier Modes §variances.

Step 1: clause 3. Fix $x\in H^{-1}$. By (P1), (P2) and claim 5 of Elementary Arithmetic in an Ordered Field (multiplying $0\le q_{k}\le\frac{\beta}{2\mu_{k}}$ by $x(k)^{2}\ge0$), $0\le q_{k}x(k)^{2}\le\frac{\beta}{2}\cdot\frac{x(k)^{2}}{\mu_{k}}$, so $|q_{k}x(k)^{2}|\le\frac{\beta}{2}\cdot\frac{x(k)^{2}}{\mu_{k}}$ by (P1). The family $k\mapsto x(k)^{2}/\mu_{k}$ is cube-summable by The Wick-Square Problem on the Torus: Standing Notation §state-space, hence so is its multiple by $\beta/2$ (Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear), and $k\mapsto q_{k}x(k)^{2}$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §comparison. The family $k\mapsto a_{k}$ is cube-summable by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §constants, so $k\mapsto q_{k}x(k)^{2}+a_{k}$ is cube-summable by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear, with

$$R(x)=\sum_{k\in\mathbb{Z}^{n}}q_{k}x(k)^{2}+\sum_{k\in\mathbb{Z}^{n}}a_{k}.$$

By Cube Sums of Families on the Integer Lattice §cube-sums, $u_{N}(x)$ is the $N$th cube sum of this family, so by Cube Sums of Families on the Integer Lattice §lattice-sum the sequence $(u_{N}(x))_{N\in\mathbb{N}}$ converges to $R(x)$. This proves clause 3.

Step 2: mode derivatives of diagonal quadratics. Let $\rho,\alpha:\mathbb{Z}^{n}\to\mathbb{R}$ be families such that for every $y\in H^{-1}$ the family $k\mapsto\rho(k)y(k)^{2}+\alpha(k)$ is cube-summable, and let $\Phi(y)$ be its lattice sum. We show: $\Phi$ is twice differentiable along the modes, with $\partial_{k}\Phi(y)=2\rho(k)y(k)$ and $\partial_{k}^{2}\Phi(y)=2\rho(k)$ for all $y\in H^{-1}$ and $k\in\mathbb{Z}^{n}$.

Fix $y$, $k$ and $t\in\mathbb{R}$. Then $y+te_{k}\in H^{-1}$ (The Wick-Square Problem on the Torus: Standing Notation §units), with $(y+te_{k})(k')=y(k')$ for $k'\ne k$ and $(y+te_{k})(k)=y(k)+t$. Let $a$ and $b$ be the families $k'\mapsto\rho(k')y(k')^{2}+\alpha(k')$ and $k'\mapsto\rho(k')(y+te_{k})(k')^{2}+\alpha(k')$, both cube-summable by hypothesis, and $\delta=b-a$. Then $\delta(k')=0$ for $k'\ne k$, and by claim 5 of Zero Products and Elementary Identities in a Field, $\delta(k)=\rho(k)\bigl((y(k)+t)^{2}-y(k)^{2}\bigr)=2\rho(k)y(k)\,t+\rho(k)\,t^{2}$. By Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support with the set $\{k\}$, $\delta$ is cube-summable with lattice sum $\delta(k)$ (the sum over a one-point set being its single term, Sum over a Finite Index Set); since $b=a+\delta$, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear gives

$$\Phi(y+te_{k})=\Phi(y)+2\rho(k)y(k)\,t+\rho(k)\,t^{2}.$$

By the constant, power, sum and constant-multiple rules of Single-Variable Calculus on an Interval §derivative, this section is differentiable at every $t$ with derivative $2\rho(k)y(k)+2\rho(k)t$, which is differentiable at $0$ with derivative $2\rho(k)$. The claim follows from Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §twice and Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes §derivatives.

Step 3: the cutoff operator on functions with the Riccati profile. Let $N\in\mathbb{N}$ and let $\Phi:H^{-1}\to\mathbb{R}$ be twice differentiable along the modes with $\partial_{k}\Phi(y)=2q_{k}y(k)$ and $\partial_{k}^{2}\Phi(y)=2q_{k}$ for every $y\in H^{-1}$ and every $k\in\Gamma_{N}$. We show $F_{N}[\Phi](x)=\gamma\Phi(x)-\gamma u_{N}(x)$ for every $x\in H^{-1}$.

By The Free-Field Generator with a Mode Cutoff §generator, The Gradient Energy with a Mode Cutoff §gradient and The Wick Square with a Mode Cutoff and the Wick Domain §cutoff, and field arithmetic in each term,

$$L_{N}\Phi(x)=\sum_{k\in\Gamma_{N}}\bigl(\nu q_{k}-2\mu_{k}q_{k}x(k)^{2}\bigr),\qquad|D_{N}\Phi(x)|^{2}=\sum_{k\in\Gamma_{N}}4q_{k}^{2}x(k)^{2},\qquad{:}x^{2}{:}_{N}=\sum_{k\in\Gamma_{N}}\bigl(x(k)^{2}-c_{k}\bigr).$$

By The Cutoff Hamilton-Jacobi-Bellman Operators of the Wick-Square Problem, with a General Counterterm §wick with $g=0$, and claims 3 and 4 of Properties of a Sum over a Finite Index Set,

$$F_{N}[\Phi](x)=\gamma\Phi(x)+\sum_{k\in\Gamma_{N}}r(k),\qquad r(k)=-\nu q_{k}+2\mu_{k}q_{k}x(k)^{2}+2q_{k}^{2}x(k)^{2}-\beta x(k)^{2}+\beta c_{k}.$$

Regrouping and using the Riccati equation and $\gamma a_{k}=\nu q_{k}-\beta c_{k}$ from (P2),

$$r(k)=\bigl(2q_{k}^{2}+(\gamma+2\mu_{k})q_{k}-\beta\bigr)x(k)^{2}-\gamma q_{k}x(k)^{2}-(\nu q_{k}-\beta c_{k})=-\gamma\bigl(q_{k}x(k)^{2}+a_{k}\bigr).$$

By claim 4 of Properties of a Sum over a Finite Index Set, $\sum_{k\in\Gamma_{N}}r(k)=-\gamma u_{N}(x)$, which proves the claim.

Clause 1. Fix $N$. Let $\rho_{N}(k)=q_{k}$ and $\alpha_{N}(k)=a_{k}$ for $k\in\Gamma_{N}$, and $\rho_{N}(k)=\alpha_{N}(k)=0$ otherwise. For $y\in H^{-1}$, the family $k\mapsto\rho_{N}(k)y(k)^{2}+\alpha_{N}(k)$ vanishes off $\Gamma_{N}$ (claim 1 of Zero Products and Elementary Identities in a Field), and $\Gamma_{N}$ is a nonempty finite set (The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points §finite); by Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §finite-support it is cube-summable with lattice sum $\sum_{k\in\Gamma_{N}}\bigl(q_{k}y(k)^{2}+a_{k}\bigr)=u_{N}(y)$. By Step 2, $u_{N}$ is twice differentiable along the modes with $\partial_{k}u_{N}(y)=2\rho_{N}(k)y(k)$ and $\partial_{k}^{2}u_{N}(y)=2\rho_{N}(k)$; for $k\in\Gamma_{N}$ these are $2q_{k}y(k)$ and $2q_{k}$. By Step 3, $F_{N}[u_{N}](x)=\gamma u_{N}(x)-\gamma u_{N}(x)=0$ for every $x\in H^{-1}$, which is the equation of The Cutoff Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem, Wick-Ordered and Bare §wick.

Clause 2. Fix $N$. For $x\in H^{-1}$ and $k\in\mathbb{Z}^{n}$ the section of $v_{N}$ at $x$ along $k$ is the section of $u_{N}$ plus the constant function with value $C_{N}\gamma^{-1}$. By the constant and sum rules of Single-Variable Calculus on an Interval §derivative and clause 1, it is differentiable everywhere with the same derivative as the section of $u_{N}$, which is differentiable at $0$. So $v_{N}$ is twice differentiable along the modes with $\partial_{k}v_{N}=\partial_{k}u_{N}$ and $\partial_{k}^{2}v_{N}=\partial_{k}^{2}u_{N}$, and hence $L_{N}v_{N}(x)=L_{N}u_{N}(x)$ and $|D_{N}v_{N}(x)|^{2}=|D_{N}u_{N}(x)|^{2}$ (The Free-Field Generator with a Mode Cutoff §generator, The Gradient Energy with a Mode Cutoff §gradient). By claims 3 and 4 of Properties of a Sum over a Finite Index Set, $\beta\,{:}x^{2}{:}_{N}=\beta\sum_{k\in\Gamma_{N}}x(k)^{2}-C_{N}$. Therefore, by The Cutoff Hamilton-Jacobi-Bellman Operators of the Wick-Square Problem, with a General Counterterm §bare and The Cutoff Hamilton-Jacobi-Bellman Operators of the Wick-Square Problem, with a General Counterterm §wick with $g=0$,

$$F_{N}^{0}[v_{N}](x)=\gamma u_{N}(x)+C_{N}-L_{N}u_{N}(x)+\tfrac12|D_{N}u_{N}(x)|^{2}-\beta\sum_{k\in\Gamma_{N}}x(k)^{2}=F_{N}[u_{N}](x)=0$$

by clause 1; this is the equation of The Cutoff Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem, Wick-Ordered and Bare §bare.

Now let $2\le n$ and $x\in H^{-1}$, and suppose, for a contradiction, that the sequence $(v_{N}(x))_{N\in\mathbb{N}}$ is bounded above (The Real Numbers: Standing Notation and Background §bounds), say $v_{N}(x)\le M$ for every $N$. By Step 1 the sequence $(u_{N}(x))_{N\in\mathbb{N}}$ converges, so it is bounded (claim 2 of Uniqueness of Limits and Boundedness of Convergent Real Sequences): there is $B>0$ with $|u_{N}(x)|\le B$, hence $-B\le u_{N}(x)$ and $-u_{N}(x)\le B$, for every $N$ (Bounded Sequence of Real Numbers, claim 6 of Properties of the Absolute Value in an Ordered Field, claim 4 of Elementary Order Arithmetic in an Ordered Field). Put $s_{N}=\sum_{k\in\Gamma_{N}}\frac{1}{\mu_{k}}$ and $\kappa=\frac{\beta\nu}{2\gamma}$, which is positive by (P2). Since $c_{k}=\frac{\nu}{2}\cdot\frac{1}{\mu_{k}}$, claim 4 of Properties of a Sum over a Finite Index Set gives $C_{N}\gamma^{-1}=\kappa s_{N}$, so $\kappa s_{N}=v_{N}(x)-u_{N}(x)\le M+B$ by (P1). Multiplying by $\kappa^{-1}>0$ (claim 5 of Elementary Arithmetic in an Ordered Field), $s_{N}\le\kappa^{-1}(M+B)$ for every $N$, so the set $\{s_{N}:N\in\mathbb{N}\}$ is bounded above, contradicting Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums §divergent. Hence $(v_{N}(x))_{N\in\mathbb{N}}$ is not bounded above.

Clause 4. Put $\eta(k)=q_{k}-\frac{\beta}{2\mu_{k}}$ for $k\in\mathbb{Z}^{n}$ and $C=\frac{\beta\gamma+\beta^{2}}{4}$; $0\le C$ by (P2). By The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §profile, $\frac{\beta}{2\mu_{k}}-q_{k}$ is nonnegative and at most $\frac{\beta\gamma+\beta^{2}}{4\mu_{k}^{2}}$; since $\eta(k)$ is its negative, $|\eta(k)|=\frac{\beta}{2\mu_{k}}-q_{k}$ by claim 2 of Properties of the Absolute Value in an Ordered Field and (P1). Multiplying by $\mu_{k}^{2}\ge0$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $\mu_{k}^{2}|\eta(k)|\le C$, so $\eta$ satisfies Regular Functions for the Wick-Square Problem: Diagonal Quadratics of Curvature of Order the Inverse Squared Weight, plus Twice Differentiable Functions on the Sobolev Space of Order -3 §curvature with curvature bound $C$.

For the cylindrical part take $m=1$, the single mode $k_{1}=0$ (the mode all of whose components are $0$), and $f:\mathbb{R}^{1}\to\mathbb{R}$ the constant function with value $A=\sum_{k\in\mathbb{Z}^{n}}a_{k}$, which exists by The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants §constants. We check that $f$ is of class $C^{2}$ on $\mathbb{R}^{1}$ (C^k Maps on a Euclidean Open Set, read with its scalar convention, clause 3); $\mathbb{R}^{1}$ is open in itself by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous. Let $G:\mathbb{R}^{1}\to\mathbb{R}$ be any constant function, with value $b$. (i) $G$ is continuous at every point $z$ in the sense of Continuity at a Point for Maps Between Euclidean Spaces: for $\varepsilon>0$ take $\delta=1$; for every $y\in\mathbb{R}^{1}$ the one-term sum $\sum_{j=1}^{1}(G(y)-G(z))^{2}$ is $(b-b)^{2}=0$ (Finite Sum Notation in a Field), and $0<\varepsilon^{2}$ by claim 5 of Elementary Order Arithmetic in an Ordered Field. (ii) At every $z\in\mathbb{R}^{1}$ the partial derivative of $G$ with respect to the first variable exists with value $0$ (Partial Derivative on a Euclidean Open Set): for $\varepsilon>0$ take $\delta=1$; for $0<|h|<1$ the point $(z_{1}+h)$ lies in $\mathbb{R}^{1}$ and the difference quotient minus $0$ is $(b-b)h^{-1}=0$, of absolute value $0<\varepsilon$. The value is unique, because the defining condition is that of differentiability at $0$ of $t\mapsto G((z_{1}+t))$ in Single-Variable Calculus on an Interval §derivative, whose derivative is unique by that clause. Thus $\partial_{1}G$ is the zero function, which is constant and hence continuous at every point by (i). By clause 1 of C^k Maps on a Euclidean Open Set, every constant function on $\mathbb{R}^{1}$ is of class $C^{1}$. Applying this to $f$ and to the constant function $\partial_{1}f=0$, clause 2 of C^k Maps on a Euclidean Open Set (with $k=1$) shows that $f$ is of class $C^{2}$.

It remains to verify the representation. Fix $x\in H^{-1}$. By Step 1, $R(x)=\sum_{k}q_{k}x(k)^{2}+\sum_{k}a_{k}$; by The Gaussian Penalty of the Wick-Square Problem, $P(x)=\frac{\beta}{2}\sum_{k}\frac{x(k)^{2}}{\mu_{k}}$. Since $\eta(k)x(k)^{2}=q_{k}x(k)^{2}-\frac{\beta}{2}\cdot\frac{x(k)^{2}}{\mu_{k}}$, Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families §linear (with Step 1 and The Wick-Square Problem on the Torus: Standing Notation §state-space) shows that $k\mapsto\eta(k)x(k)^{2}$ is cube-summable with $\sum_{k}\eta(k)x(k)^{2}=\sum_{k}q_{k}x(k)^{2}-P(x)$. Hence

$$(R-P)(x)=\sum_{k\in\mathbb{Z}^{n}}\eta(k)\,x(k)^{2}+A=\sum_{k\in\mathbb{Z}^{n}}\eta(k)\,x(k)^{2}+f\bigl(x(k_{1})\bigr),$$

and $R-P$ is regular by Regular Functions for the Wick-Square Problem: Diagonal Quadratics of Curvature of Order the Inverse Squared Weight, plus Twice Differentiable Functions on the Sobolev Space of Order -3.

Clause 5. By Step 1, the hypothesis of Step 2 holds with $\rho(k)=q_{k}$ and $\alpha(k)=a_{k}$, and the resulting $\Phi$ is $R$. So $R$ is twice differentiable along the modes with $\partial_{k}R(y)=2q_{k}y(k)$ and $\partial_{k}^{2}R(y)=2q_{k}$ for all $y$ and $k$, and Step 3 gives, for every $x\in H^{-1}$ and $N\in\mathbb{N}$,

$$F_{N}[R](x)=\gamma R(x)-\gamma u_{N}(x).$$

The constant sequence $(\gamma R(x))_{N}$ converges to $\gamma R(x)$ (Constant Sequences and Index-Shifted Sequences of Real Numbers §constant), and $(\gamma u_{N}(x))_{N}$ converges to $\gamma R(x)$ by Step 1 and claim 3 of Arithmetic of Limits of Real Sequences; by claim 3 of that theorem again, $(F_{N}[R](x))_{N\in\mathbb{N}}$ converges to $\gamma R(x)-\gamma R(x)=0$. Hence $(R,x)\in\mathcal{D}$ (The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators §domain) and $F[R](x)=0$ (The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators §operator) for every $x\in H^{-1}$, which is the equation of The Renormalised Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem §equation.